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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

π“œπ“-convergence and lim-infπ“œ-convergence in partially ordered sets

Tao Sun / Qingguo Li / Nianbai Fan
Published Online: 2018-09-18 | DOI:Β https://doi.org/10.1515/math-2018-0090

Abstract

In this paper, we first introduce the notion of π“œπ“-convergence in posets as an unified form of O-convergence and O2-convergence. Then, by studying the fundamental properties of π“œπ“-topology which is determined by π“œπ“-convergence according to the standard topological approach, an equivalent characterization to the π“œπ“-convergence being topological is established. Finally, the lim-infπ“œ-convergence in posets is further investigated, and a sufficient and necessary condition for lim-infπ“œ-convergence to be topological is obtained.

Keywords: π“œπ“-convergence; π“œπ“-topology; lim-infπ“œ-convergence; π“œ-topology

MSC 2010: 54A20; 06A06

1 Introduction, Notations and Preliminaries

The concept of O-convergence in partially ordered sets (posets, for short) was introduced by Birkhoff [1], Frink [2] and Mcshane [3]. It is defined as follows: a net (xi)i∈I in a poset P is said to O-converge to x ∈ P if there exist subsets D and F of P such that

  1. D is directed and F is filtered;

  2. sup D = x = inf F;

  3. for every d ∈ D and e ∈ F, d ⩽ xi ⩽ e holds eventually, i.e., there exists i0 ∈ I such that d ⩽ xi ⩽ e for all i ⩾ i0.

As what has been showed in [4], the O-convergence (Note: in [4], the O-convergence is called order-convergence) in a general poset P may not be topological, i.e., it is possible that P can not be endowed with a topology such that the O-convergence and the associated topological convergence are consistent. Hence, much work has been done to characterize those special posets in which the O-convergence is topological. The most recent result in [5] shows that the O-convergence in a poset which satisfies Condition (β–³) is topological if and only if the poset is π“ž-doubly continuous. This means that for a special class of posets, a sufficient and necessary condition for O-convergence being topological is obtained.

As a direct generalization of O-convergence, O2-convergence in posets has been discussed in [11] from the order-theoretical point of view. It is defined as follows: a net (xi)i∈I in a poset P is said to O2-converge to x ∈ P if there exist subsets A and B of P such that

  1. sup A = x = inf B;

  2. for every a ∈ A and b ∈ B, a ⩽ xi ⩽ b holds eventually.

In fact, the O2-convergence is also not topological generally. To clarify those special posets in which the O2-convergence is topological, Zhao and Li [6] showed that for any poset P satisfying Condition (βˆ—), O2-convergence is topological if and only if P is Ξ±-doubly continuous. As a further result, Li and Zou [7] proved that the O2-convergence in a poset P is topological if and only if P is O2-doubly continuous. This result demonstrates the equivalence between the O2-convergence being topological and the O2-double continuity of a given poset.

On the other hand, Zhou and Zhao [8] have defined the lim-infπ“œ-convergence in posets to generalize lim-inf-convergence and lim-inf2-convergence [4]. They also found that the lim-infπ“œ-convergence in a poset is topological if and only if the poset is Ξ±(π“œ)-continuous when some additional conditions are satisfied (see [8], Theorem 3.1). This result clarified some special conditions of posets under which the lim-infπ“œ-convergence is topological. However, to the best of our knowledge, the equivalent characterization to the lim-infπ“œ-convergence in general posets being topological is still unknown.

One goal of this paper is to propose the notion of π“œπ“-convergence in posets which can unify O-convergence and O2-convergence and search the equivalent characterization to the π“œπ“-convergence being topological. More precisely,

  • (G11)

    Given a general poset P, we hope to clarify the order-theoretical condition of P which is sufficient and necessary for the π“œπ“-convergence being topological.

  • (G12)

    Given a poset P satisfying such condition, we hope to provide a topology that can be equipped on P such that the π“œπ“-convergence and the associated topological convergence agree.

Another goal is to look for the equivalent characterization to the lim-infπ“œ-convergence being topological. More precisely,

  • (G21)

    Given a general poset P, we expect to present a sufficient and necessary condition of P which can precisely serve as an order-theoretical condition for the lim-infπ“œ-convergence being topological.

  • (G22)

    Given a poset P satisfying such condition, we expect to give a topology on P such that the lim-infπ“œ-convergence and the associated topological convergence are consistent.

To accomplish those goals, motivated by the ideal of introducing the Z-subsets system [9] for defining Z-continuous posets, we propose the notion of π“œπ“-doubly continuous posets and define the π“œπ“-topology on posets in Section 2. Based on the study of the basic properties of the π“œπ“-topology, it is proved that the π“œπ“-convergence in a poset P is topological if and only if P is an π“œπ“-doubly continuous poset if and only if the π“œπ“-convergence and the topological convergence with respect to π“œπ“-topology are consistent. In Section 3, by introducing the notion of Ξ±*(π“œ)-continuous posets and presenting the fundamental properties of π“œ-topology which is induced by the lim-infπ“œ-convergence, we show that the lim-infπ“œ-convergence in a poset P is topological if and only if P is an Ξ±*(π“œ)-continuous poset if and only if the lim-infπ“œ-convergence and the topological convergence with respect to π“œ-topology are consistent.

Some conventional notations will be used in the paper. Given a setX, F βŠ‘ X means that F is a finite subset of X. Given a topological space (X, 𝓣) and a net (xi)i∈I in X, we take (xi)i∈Iβ†’Tx to mean the net (xi)i∈I converges to x ∈ P with respect to the topology 𝓣.

Let P be a poset and x ∈ P. ↑ x and ↓ x are always used to denote the principal filter {y ∈ P : y β©Ύ x} and the principal ideal {z ∈ P : z β©½ x} of P, respectively. Given a poset P and A βŠ† P, by writing sup A we mean that the least upper bound of A in P exists and equals to sup A ∈ P; dually, by writing inf A we mean that the greatest lower bound of A in P exists and equals to inf A ∈ P. And the set A is called an upper set if A = ↑A = {b ∈ P; (βˆƒa ∈ A) a β©½ b}, the lower set is defined dually.

For a poset P, we succinctly denote

  • –

    π“Ÿ(P) = {A : A βŠ† P}; π“Ÿ0(P) = π“Ÿ(P)/{βˆ…};

  • –

    𝓓(P) = {D ∈ π“Ÿ(P): D is a directed subset of P};

  • –

    𝓕(P) = {F ∈ π“Ÿ(P): F is a filtered subset of P};

  • –

    𝓛(P) = {L ∈ π“Ÿ(P): L βŠ‘ P}; 𝓛0(P) = 𝓛(P)/{βˆ…};

  • –

    𝓒0(P) = {{x} : x ∈ P}.

To make this paper self-contained, we briefly review the following notions:

Definition 1.1

([5]). Let P be a poset and x, y, z ∈ P. We say yβ‰ͺπ“žx if for every net (xi)i∈I in P which O-converges to x ∈ P, xi β©Ύ y holds eventually; dually, we say zβŠ²π“žx if for every net (xi)i∈I in P which O-converges to x ∈ P, xi β©½ z holds eventually.

Definition 1.2

([5]). A poset P is said to be π“ž-doubly continuous if for every x ∈ P, the set {a ∈ P : aβ‰ͺπ“žx} is directed, the set {b ∈ P : bβŠ²π“žx} is filtered and sup{a ∈ P : aβ‰ͺπ“žx} = x = inf{b ∈ P : bβŠ²π“žx}.

Condition (β–³). A poset P is said to satisfy Condition(β–³) if

  1. for any x, y, z ∈ P, xβ‰ͺπ“žy β©½ z implies xβ‰ͺπ“žz;

  2. for any w, s, t ∈ P, wβŠ²π“žs β©Ύ t implies wβŠ²π“žt.

Definition 1.3

([6]). Let P be a poset and x, y, z ∈ P. We say yβ‰ͺΞ±x if for every net (xi)i∈I in P which O2-converges to x ∈ P, xi β©Ύ y holds eventually; dually, we say z⊲αx if for every net (xi)i∈I in P which O2-converges to x ∈ P, xi β©½ z holds eventually.

Definition 1.4

([7]). A poset P is said to be O2-doubly continuous if for every x ∈ P,

  1. sup{a ∈ P : aβ‰ͺΞ±x} = x = inf{b ∈ P : b⊲αx};

  2. for any y, z ∈ P with yβ‰ͺΞ±x and z⊲αx, there exist A βŠ‘ {a ∈ P : aβ‰ͺΞ±x} and B βŠ‘ {b ∈ P : b⊲αx} such that yβ‰ͺΞ±c and z⊲αc for each c ∈ β‹‚{↑a ∩ ↓b: a ∈ A & b ∈ B}.

2 π“œπ“-topology on posets

Based on the introduction of π“œπ“-convergence in posets, the π“œπ“-topology can be defined on posets. In this section, we first define the π“œπ“-double continuity for posets. Then, we show the equivalence between the π“œπ“-convergence being topological and the π“œπ“-double continuity of a given poset.

A PMN-space is a triplet (P, π“œ,𝓝) which consists of a poset P and two subfamily π“œ,𝓝 βŠ† π“Ÿ(P).

All PMN-spaces (P, π“œ,𝓝) considered in this section are assumed to satisfy the following conditions:

  • (C1)

    If P has the least element βŠ₯, then {βŠ₯} ∈ π“œ;

  • (C2)

    If P has the greatest element ⊀, then {⊀} ∈ 𝓝;

  • (C3)

    βˆ… βˆ‰ π“œ and βˆ… βˆ‰ 𝓝.

Definition 2.1

Let (P, π“œ,𝓝) be a PMN-space. A net (xi)i∈I in P is said to π“œπ“-converge to x ∈ P if there exist M ∈ π“œ and N ∈ 𝓝 satisfying:

  • (MN1)

    sup M = x = inf N;

  • (MN2)

    xi ∈ ↑m ∩ ↓n eventually for every m ∈ M and every n ∈ N.

In this case, we will write (xi)i∈Iβ†’MNx.

Remark 2.2

Let (P, π“œ,𝓝) be a PMN-space.

  1. If π“œ = 𝓓(P) and 𝓝 = 𝓕(P), then a net (xi)i∈Iβ†’MNx ∈ P if and only if it O-converges to x. That is to say, O-convergence is a particular case of π“œπ“-convergence.

  2. If π“œ = 𝓝 = π“Ÿ0(P), then a net (xi)i∈Iβ†’MNx ∈ P if and only if it O2-converges to x. That is to say, O2-convergence is a special case of π“œπ“-convergence.

  3. If π“œ = 𝓝 = 𝓛0(P), then a net (xi)i∈Iβ†’MNx ∈ P if and only if xi = x holds eventually.

  4. The π“œπ“-convergent point of a net (xi)i∈I in P, if exists, is unique.

    Indeed, suppose that (xi)i∈Iβ†’MNx1 and (xi)i∈Iβ†’MNx2. Then there exist Ak ∈ π“œ and Bk ∈ 𝓝 such that sup Ak = xk = inf Bk and ak β©½ xi β©½ bk holds eventually for every ak ∈ Ak and bk ∈ Bk (k = 1, 2). This implies that for any a1 ∈ A1, a2 ∈ A2, b1 ∈ B1 and b2 ∈ B2, there exists i0 ∈ I such that a1 β©½ xi0 β©½ b2 and a2 β©½ xi0 β©½ b1. Thus we have sup A1 = x1 β©½ inf B2 = x2 and sup A2 = x2 β©½ inf B1 = x1. Therefore x1 = x2.

  5. For any A ∈ π“œ and B ∈ 𝓝 with sup A = inf B = x ∈ P, we denote F(A,B)x = {β‹‚{↑a ∩ ↓b : a ∈ A0 & b ∈ B0}: A0 βŠ‘ A & B0 βŠ‘ B}1. Let D(A,B)x={(d,D)∈PΓ—F(A,B)x:d∈D} and let the preorder ≀ on D(A,B)x be defined by

    (βˆ€(d1,D1),(d2,D2)∈D(A,B)x)(d1,D1)≀(d2,D2)⟺D2βŠ†D1.

    One can readily check that (D(A,B)x,≀) is directed. Now if we take x(d,D) = d for every (d,D)∈D(A,B)x,, then the net (x(d,D))(d,D)∈D(A,B)xβ†’MNx because sup A = inf B = x, and a β©½ x(d,D) β©½ b holds eventually for any a ∈ A and b ∈ B.

  6. Let (x(d,D))(d,D)∈D(A,B)x be the net defined in (5) for any A ∈ π“œ and B ∈ 𝓝 with sup A = inf B = x ∈ P. If (x(d,D))(d,D)∈D(A,B)x converges to p ∈ P with respect to some topology 𝓣 on the poset P, then for every open neighborhood Up of p, there exist A0 βŠ‘ A and B0 βŠ‘ B such that

    β‹‚{↑aβˆ©β†“b:a∈A0&b∈B0}βŠ†Up.

    Indeed, suppose that (x(d,D))(d,D)∈D(A,B)xβ†’Tp. Then for every open neighborhood Up of p, there exists (d0,D0) ∈ D(A,B)x such that x(d,D) = d ∈ Up for all (d, D) β‰₯ (d0,D0). Since (d, D0) β‰₯ (d0,D0) for every d ∈ D0, x(d,D) = d ∈ Up for every d ∈ D0. This shows D0 βŠ† Up. So, there exist A0 βŠ‘ A and B0 βŠ‘ B such that

    D0=β‹‚{↑aβˆ©β†“b:a∈A0&b∈B0}βŠ†Up.

Given a PMN-space (P, π“œ,𝓝), we can define two new approximate relations β‰ͺMN and ⊳MN on the poset P in the following definition.

Definition 2.3

Let (P, π“œ,𝓝) be a PMN-space and x, y, z ∈ P.

  1. We define yβ‰ͺMNx if for any A ∈ π“œ and B ∈ 𝓝 with sup A = x = inf B, there exist A0 βŠ‘ A and B0 βŠ‘ B such that

    β‹‚{↑aβˆ©β†“b:a∈A0&b∈B0}βŠ†β†‘y.

  2. Dually, we define z⊳MNx if for any M ∈ π“œ and N ∈ 𝓝 with sup M = x = inf N, there exist M0 βŠ‘ M and N0 βŠ‘ N such that

    β‹‚{↑mβˆ©β†“n:m∈M0&n∈N0}βŠ†β†“z.

For convenience, given a PMN-space (P, π“œ,𝓝) and x ∈ P, we will briefly denote

  • –

    β–ΎMNx={y∈P:yβ‰ͺMNx};

  • –

    β–΄MNx={z∈P:xβ‰ͺMNz};

  • –

    β–½MNx={a∈P:x⊳MNa};

  • –

    β–³MNx={b∈P:b⊳MNx}.

Remark 2.4

Let (P, π“œ,𝓝) be a PMN-space and x, y, z ∈ P.

  1. If there is no A ∈ π“œ such that sup A = x, then pβ‰ͺMNx and p⊳MNx for all p ∈ P; similarly, if there is no B ∈ 𝓝 such that inf B = x, then pβ‰ͺMNx and p⊳MNx for all p ∈ P.

  2. By Definition 2.3, one can easily check that if P has the least element βŠ₯, then βŠ₯β‰ͺMNp for every p ∈ P, and if P has the greatest element ⊀, then ⊀⊳MNp for every p ∈ P.

  3. The implications yβ‰ͺMNxβ‡’xβ©½yandz⊳MNxβ‡’zβ©Ύx are not true necessarily. See the following example: let ℝ be the set of all real numbers, in its ordinal order, and π“œ = 𝓝 = {{n} : n ∈ β„€}, where β„€ is the set of all integers. Then, by (1), we have 1β‰ͺMN1/2and0⊳MN1/2. But 1β§Έ β©½ 1/2 and 0β§Έ β©Ύ 1/2.

  4. Assume that sup A0 = x = inf B0 for some A0 ∈ π“œ and B0 ∈ 𝓝. Then it follows from Definition 2.3 that yβ‰ͺMNx implies y β©½ x and z⊳MNx implies z β©Ύ x. In particular, if 𝓒0(P) βŠ† π“œ,𝓝, then bβ‰ͺMNa implies b β©½ a and c⊳MNa implies c β©Ύ a for any a, b, c ∈ P. More particularly, for any p1,p2,p3 ∈ P, we have p1β‰ͺS0S0 p2 ⟺ p1 β©½ p2 and p3⊳S0S0p2 ⟺ p3 β©Ύ p2.

Proposition 2.5

Let (P, π“œ,𝓝) be a PMN-space and x, y, z ∈ P. Then

  1. yβ‰ͺMNx if and only if for every net (xi)i∈I that π“œπ“-converges to x, xi β©Ύ y holds eventually.

  2. z⊳MNx if and only if for every net (xi)i∈I that π“œπ“-converges to x, xi β©½ z holds eventually.

Proof

(1) Suppose yβ‰ͺMNx. If a net (xi)i∈Iβ†’MNx, then there exist A ∈ π“œ and B ∈ 𝓝 such that sup A = x = inf B, and for any a ∈ A and b ∈ B, there exists iab∈I such that a β©½ xi β©½ b for all iβ©Ύiab. According to Definition 2.3 (1), it follows that there exist A0 = {a1,a2, …,an} βŠ‘ A and B0 = {b1,b2, …,bm} βŠ‘ B such that x ∈ β‹‚{↑ak ∩ ↓bj : 1 ≀ k ≀ n & 1 ≀ j ≀ m} βŠ† ↑y. Take i0 ∈ I with that i0β©Ύiakbj for every k ∈ {1, 2, …, n} and every j ∈ {1, 2, …, m}. Then xi ∈ β‹‚{↑ak ∩ ↓bj : 1 ≀ k ≀ n & 1 ≀ j ≀ m} βŠ† ↑y for all i β©Ύ i0. This means xi β©Ύ y holds eventually.

Conversely, suppose that for every net (xi)i∈I that π“œπ“-converges to x, xi β©Ύ y holds eventually. For every A ∈ π“œ and B ∈ 𝓝 with sup A = x = inf B, consider the net (x(d,D))(d,D)∈D(A,B)x defined in Remark 2.2 (5). By Remark 2.2 (5), the net (x(d,D))(d,D)∈D(A,B)xβ†’MNx. So, there exists (d0,D0) ∈ D(A,B)x such that x(d,D) = d β©Ύ y for all (d, D) β‰₯ (d0,D0). Since (d, D0) β‰₯ (d0,D0) for all d ∈ D0, x(d,D0) = d β©Ύ y for all d ∈ D0. Thus, we have D0 βŠ† ↑y. It follows from the definition of D(A,B)x that there exist A0 βŠ‘ A and B0 βŠ‘ B such that D0 = β‹‚{↑a ∩ ↓b : a ∈ A0 & b ∈ B0} βŠ† ↑y. This shows yβ‰ͺMNx.

The proof of (2) can be processed similarly. ░

Remark 2.6

Let (P, π“œ,𝓝) be a PMN-space.

  1. If π“œ = 𝓓(P) and 𝓝 = 𝓕(P), then β‰ͺDF=β‰ͺOand⊳DF=⊳O.

  2. If π“œ = 𝓝 = π“Ÿ0(P), then β‰ͺP0P0=β‰ͺΞ±and⊳P0P0=⊳α.

Given a PMN-space (P, π“œ,𝓝), depending on the approximate relations β‰ͺMNand⊳MN on P. we can define the π“œπ“-double continuity for the poset P.

Definition 2.7

Let (P, π“œ,𝓝) be a PMN-space. The poset P is called an π“œπ“-doubly continuous poset if for every x ∈ P, there exist Mx ∈ π“œ and Nx ∈ 𝓝 such that

  • (A1)

    Mx βŠ† β–ΎMNx,NxβŠ†β–³MNx and sup Mx = x = inf Nx.

  • (A2)

    For any yβˆˆβ–ΎMNxandzβˆˆβ–³MNx, β‹‚{↑m ∩ ↓n : m ∈ M0 & n ∈ N0} βŠ† β–΄MNyβˆ©β–½MNz for some M0 βŠ‘ Mx and N0 βŠ‘ Nx.

By Remark 2.4 (4) and Definition 2.7, we have the following basic property about π“œπ“-doubly continuous posets:

Proposition 2.8

Let (P, π“œ,𝓝) be a PMN-space and x, y, z ∈ P. If the poset P is an π“œπ“-doubly continuous poset, then yβ‰ͺMNx implies y β©½ x and z⊳MNx implies z β©Ύ x.

Example 2.9

Let (P, π“œ,𝓝) be a PMN-space.

  1. If π“œ = 𝓝 = 𝓒0(P), then by Remark 2.4 (4), we have β‰ͺS0S0=β©½and⊳S0S0=β©Ύ. By Definition 2.7, one can easily check that P is an 𝓒0𝓒0-doubly continuous poset.

  2. If π“œ = 𝓝 = 𝓛0(P), then by Definition 2.3, we have β‰ͺL0L0=β©½and⊳L0L0=β©Ύ. It can be easily checked from Definition 2.7 that P is an 𝓛0𝓛0-doubly continuous poset.

  3. Let π“œ = 𝓓(P) and 𝓝 = 𝓕(P). Then it is easy to check that if P is an π“ž-doubly continuous poset which satisfies Condition (β–³), then it is a 𝓓𝓕-doubly continuous poset. Particularly, finite posets, chains and anti-chains, completely distributive lattices are all 𝓓𝓕-doubly continuous posets.

  4. Let π“œ = 𝓝 = π“Ÿ0(P). Then the poset P is π“Ÿ0π“Ÿ0-double continuous if and only if it is O2-double continuous. Thus, chains and finite posets are all π“Ÿ0π“Ÿ0-doubly continuous posets.

Next, we are going to consider the π“œπ“-topology on posets, which is induced by the π“œπ“-convergence.

Definition 2.10

Given a PMN-space (P, π“œ,𝓝), a subset U of P is called an π“œπ“-open set if for every net (xi)i∈I with that (xi)i∈Iβ†’MNx ∈ U, xi ∈ U holds eventually.

Clearly, the family OMN(P) consisting of all π“œπ“-open subsets of P forms a topology on P. And this topology is called the π“œπ“-topology.

Theorem 2.11

Let (P, π“œ,𝓝) be a PMN-space. Then a subset U of P is an π“œπ“-open set if and only if for every M ∈ π“œ and N ∈ 𝓝 with sup M = x = inf N ∈ U, we have

β‹‚{↑mβˆ©β†“n:m∈M0&n∈N0}βŠ†U

for some M0 βŠ‘ M and N0 βŠ‘ N.

Proof

Suppose that U is an π“œπ“-open subset of P. For every M ∈ π“œ and N ∈ 𝓝 with sup M = x = inf N ∈ U, let (x(d,D))(d,D)∈D(M,N)x be the net defined in Remark 2.2 (5). Then the net (x(d,D))(d,D)∈D(M,N)xβ†’MNx. By the definition of π“œπ“-open set, the exists (d0,D0) ∈ D(M,N)x such that x(d,D) = d ∈ U for all (d, D) β‰₯ (d0,D0). Since (d, D0) β‰₯ (d0,D0) for all d ∈ D0, x(d,D0) = d ∈ U for every d ∈ D0, and thus D0 βŠ† U. It follows from the definition of the directed set D(M,N)x that D0 = β‹‚{↑m ∩ ↓n : m ∈ M0 & n ∈ N0} βŠ† U for some M0 βŠ‘ M and some N0 βŠ‘ N.

Conversely, assume that U is a subset of P with the property that for any M ∈ π“œ and N ∈ 𝓝 with sup M = x = inf N ∈ U, there exist M0 = {m1,m2, …,mk} βŠ‘ M and N0 = {n1,n2, …,nl} βŠ‘ N such that β‹‚{↑mh ∩ ↓nj : 1 ≀ h ≀ k & 1 ≀ j ≀ l} βŠ† U. Let (xi)i∈I be a net that π“œπ“-converges to x ∈ U. Then there exist M ∈ π“œ and N ∈ 𝓝 such that sup M = x = inf N ∈ U, and for every m ∈ M and n ∈ N, m β©½ xi β©½ n holds eventually. This means that for every mh ∈ M0 and nj ∈ N0, there exists ih,j ∈ I such that mh β©½ xi β©½ nj for all i β‰₯ ih,j. Take i0 ∈ I such that i0 β‰₯ ih,j for all h ∈ {1, 2, …, k} and j ∈ {1, 2, …, l}. Then xi ∈ β‹‚{↑mh ∩ ↓nj : 1 ≀ h ≀ k & 1 ≀ j ≀ l} βŠ† U for all i β‰₯ i0. Therefore, U is an π“œπ“-open subset of P. ░

Proposition 2.12

Let (P, π“œ,𝓝) be a PMN-space in which P is an π“œπ“-doubly continuous poset, and y, z ∈ P. Then β–΄MNyβˆ©β–½MNz∈OMN(P).

Proof

Suppose that M ∈ π“œ and N ∈ 𝓝 with sup M = inf N = x ∈ β–΄MNyβˆ©β–½MNz. Since P is an π“œπ“-doubly continuous poset, there exist Mx ∈ π“œ and Nx ∈ 𝓝 satisfying condition (A1) and (A2) in Definition 2.7. This means that there exist M0 βŠ‘ Mx βŠ† β–ΎMNx and N0 βŠ‘ Nx βŠ† β–³MNx such that β‹‚{↑m0 ∩ ↓n0:m0 ∈ M0 & n0 ∈ N0} βŠ† β–΄MNyβˆ©β–½MNz. As M0 βŠ‘ Mx βŠ† β–ΎMNx and N0 βŠ‘ Nx βŠ† β–³MNx, by Definition 2.3, there exist Mm0 βŠ‘ M and Nn0 βŠ‘ N such that β‹‚{↑m ∩ ↓n : m ∈ Mm0 & n ∈ Nn0 } βŠ† ↑m0 ∩ ↓n0 for every m0 ∈ M0 and n0 ∈ N0. Take MF = ⋃{Mm0:m0 ∈ M0} and NF = ⋃{Nn0:n0 ∈ N0}. Then it is easy to check that MF βŠ‘ M, NF βŠ‘ N and

xβˆˆβ‹‚{↑aβˆ©β†“b:a∈MF&b∈NF}βŠ†β‹‚{↑m0βˆ©β†“n0:m0∈M0&n0∈N0}βŠ†β–΄MNyβˆ©β–½MNz.

So, it follows from Theorem 2.11 that β–΄MNyβˆ©β–½MNz∈OMN(P). ░

Lemma 2.13

Let (P, π“œ,𝓝) be a PMN-space in which P is an π“œπ“-doubly continuous poset. Then a net

(xi)i∈Iβ†’MNx∈P⟺(xi)i∈Iβ†’OMN(P)x.

Proof

From the definition of OMN(P), it is easy to see that a net

(xi)i∈Iβ†’MNx∈P⟹(xi)i∈Iβ†’OMN(P)x.

To prove the Lemma, it suffices to show that a net (xi)i∈Iβ†’OMN(P)x ∈ P implies (xi)i∈Iβ†’MNx. Suppose a net (xi)i∈Iβ†’OMN(P)x. Since P is an π“œπ“-doubly continuous poset, there exist Mx ∈ π“œ and Nx ∈ 𝓝 such that Mx βŠ† β–ΎMNx, Nx βŠ† β–³MNx and sup Mx = x = inf Nx. By Proposition 2.12, xβˆˆβ–΄MNyβˆ©β–½MNz∈OMN(P) for every y ∈ Mx βŠ† β–ΎMNx and every z ∈ Nx βŠ† β–³MNx, and hence xiβˆˆβ–΄MNyβˆ©β–½MNz holds eventually for every y ∈ Mx βŠ† β–ΎMNx and every z ∈ Nx βŠ† β–³MNx. It follows from Proposition 2.8 that y β©½ xi β©½ z holds eventually for every y ∈ Mx and z ∈ Nx. Thus (xi)i∈Iβ†’MNx ░

Lemma 2.14

Let (P, π“œ,𝓝) be a PMN-space. If the π“œπ“-convergence in P is topological, then P is π“œπ“-doubly continuous.

Proof

Suppose that the π“œπ“-convergence in P is topological. Then there exists a topology 𝓣 on P such that for every x ∈ P, a net (xi)i∈Iβ†’MNx if and only if (xi)i∈Iβ†’Tx. Define Ix = {(p, U) ∈ P Γ— 𝓝(x) : p ∈ U}, where 𝓝(x) denotes the set of all open neighbourhoods of x in the topological space (P, 𝓣), i.e., 𝓝(x) = {U ∈ 𝓣 : x ∈ U}. Define the preorder β‰Ό on Ix as follows:

(βˆ€(p1,U1),(p2,U2)∈Ix)(p1,U1)β‰Ό(p2,U2)⟺U2βŠ†U1.

Now one can easily see that Ix is directed. Let x(p,U) = p for every (p, U) ∈ Ix. Then it is straightforward to check that the net (x(p,U))(p,U)∈Ixβ†’Tx, and thus (x(p,U))(p,U)∈Ixβ†’MNx. By Definition 2.1, there exist Mx ∈ π“œ and Nx ∈ 𝓝 such that sup Mx = x = inf Nx, and for every m ∈ Mx and n ∈ Nx, there exists (pmn,Umn) ∈ Ix such that x(p,U) = p ∈ ↑m ∩ ↓n for all (p,U)≽(pmn,Umn). Since (p,Umn)≽(pmn,Umn) for every p∈Umn, x(p,Umn)=p ∈ ↑m ∩ ↓n for every p∈Umn. This shows

(βˆ€m∈Mx,n∈Nx)(βˆƒUmn∈N(x))x∈UmnβŠ†β†‘mβˆ©β†“n.(*)

For any A ∈ π“œ and B ∈ 𝓝 with sup A = x = inf B, let (x(d,D))(d,D)∈D(A,B)x be the net defined as in Remark 2.2 (5). Then (x(d,D))(d,D)∈D(A,B)xβ†’MNx, and hence (x(d,D))(d,D)∈D(A,B)xβ†’Tx. This implies, by Remark 2.2 (6), that there exist A0 βŠ‘ A and B0 βŠ‘ B satisfying

xβˆˆβ‹‚{↑aβˆ©β†“b:a∈A0&b∈B0}βŠ†UmnβŠ†β†‘mβˆ©β†“n.

Therefore, m ∈ β–ΎMNx and n ∈ β–³MNx, and hence β–ΎMNx and Nx βŠ† β–³MNx.

Let y ∈ β–ΎMNx and z ∈ β–³MNx. Since sup Mx = x = inf Nx, by Definition 2.3, β‹‚{↑m ∩ ↓n : m ∈ M1 & n ∈ N1} βŠ† ↑y ∩ ↓z for some M1 βŠ‘ Mx and N1 βŠ‘ Nx. This concludes by Condition (⋆) and the finiteness of sets M1 and N1 that β‹‚{Umn:m∈M1&n∈N1} ∈ 𝓝(x) and

xβˆˆβ‹‚{Umn:m∈M1&n∈N1}βŠ†β‹‚{↑mβˆ©β†“n:m∈M1&n∈N1}βŠ†β†‘yβˆ©β†“z.

Considering the net (x(d,D))(d,D)∈D(Mx,Nx)x defined in Remark 2.2 (5), we have (x(d,D))(d,D)∈D(Mx,Nx)xβ†’MNx, and hence (x(d,D))(d,D)∈D(Mx,Nx)xβ†’Tx. So, by Remark 2.2 (6), there exist M2 βŠ‘ Mx and N2 βŠ‘ Nx such that

xβˆˆβ‹‚{↑mβˆ©β†“n:m∈M2&n∈N2}βŠ†β‹‚{Umn:m∈M1&n∈N1}βŠ†β†‘yβˆ©β†“z.

Finally, we show β‹‚{↑m ∩ ↓n : m ∈ M2 & n ∈ N2} βŠ† β–΄MNyβˆ©β–½MNz. Let (x(d,D))(d,D)∈D(Mβ€²,Nβ€²)xβ€² be the net defined in 2.2 (5) for any Mβ€² ∈ π“œ and Nβ€² ∈ 𝓝 with sup Mβ€² = inf Nβ€² = xβ€² ∈ β‹‚{↑m ∩ ↓n : m ∈ M2 & n ∈ N2}. Then (x(d,D))(d,D)∈D(Mβ€²,Nβ€²)xβ€²β†’MNxβ€², and thus (x(d,D))(d,D)∈D(Mβ€²,Nβ€²)xβ€²β†’Txβ€². This implies by Remark 2.2 (6) that there exist M0β€²βŠ‘Mβ€² and N0β€²βŠ‘Nβ€² satisfying

xβ€²βˆˆβ‹‚{↑mβ€²βˆ©β†“nβ€²:m∈M0β€²&n∈N0β€²}βŠ†β‹‚{Umn:m∈M1&n∈N1}βŠ†β†‘yβˆ©β†“z.

Hence, we have xβ€² ∈ β–΄MNyβˆ©β–½MNz by Definition 2.3. This shows β‹‚{↑m ∩ ↓n : m ∈ M2 & n ∈ N2} βŠ† β–΄MNyβˆ©β–½MNz. Therefore, it follows from Definition 2.7 that P is π“œπ“-doubly continuous. ░

Combining Lemma 2.13 and Lemma 2.14, we obtain the following theorem.

Theorem 2.15

Let (P, π“œ,𝓝) be a PMN-space. Then the following statements are equivalent:

  1. P is an π“œπ“-doubly continuous poset.

  2. For any net (xi)i∈I in P, (xi)i∈Iβ†’MNx if and only if (xi)i∈Iβ†’OMN(P)x.

  3. The π“œπ“-convergence in P is topological.

Proof

(1) β‡’ (2): By Lemma 2.13.

(2) β‡’ (3): It is clear.

(3) β‡’ (1): By Lemma 2.14. ░

3 π“œ-topology induced by lim-infπ“œ-convergence

In this section, the notion of lim-infπ“œ-convergence is reviewed and the π“œ-topology on posets is defined. By exploring the fundamental properties of the π“œ-topology, those posets under which the lim-infπ“œ-convergence is topological are precisely characterized.

By saying a PM-space, we mean a pair (P, π“œ) that contains a poset P and a subfamily π“œ of π“Ÿ(P).

Definition 3.1

([8]). Let (P, π“œ) be a PM-space. A net (xi)i∈I in P is said to lim-infπ“œ-converge to x ∈ P if there exists M ∈ π“œ such that

  • (M1)

    x β©½ sup M;

  • (M2)

    for every m ∈ M, xi ⩾ m holds eventually.

In this case, we write (xi)i∈Iβ†’Mx.

It is worth noting that both lim-inf-convergence and lim-inf2-convergence [4] in posets are particular cases of lim-infπ“œ-convergence.

Remark 3.2

Let (P, π“œ) be a PM-space and x, y ∈ P.

  1. Suppose that a net (xi)i∈Iβ†’Mx and y β©½ x. (xi)i∈Iβ†’My by Definition 3.1. This concludes that the set of all lim-infπ“œ-convergent points of the net (xi)i∈I in P is a lower subset of P. Thus, the lim-infπ“œ-convergent points of the net (xi)i∈I need not be unique.

  2. If P has the least element βŠ₯ and βˆ… ∈ π“œ, then we have (xi)i∈Iβ†’MβŠ₯ for every net (xi)i∈I in P.

  3. For every M ∈ π“œ with sup M β©Ύ x, we denote FMx = {β‹‚{↑m : m ∈ M0} : M0 βŠ‘ M}2. Let DMx = {(d, D) ∈ P Γ— FMx : d ∈ D} be in the preorder ≀ defined by

    (βˆ€(d1,D1),(d2,D2)∈DMx)(d1,D1)≦(d2,D2)⟺D2βŠ†D1.

    It is easy to see that the set DMx is directed. Take x(d,D) = d for every (d, D) ∈ DMx. Then, by Definition 3.1, one can straightforwardly check that the net (x(d,D))(d,D)∈DMxβ†’M for every a β©½ x.

  4. If the net (x(d,D))(d,D)∈DMx defined in (3) converges to p ∈ P with respect to some topology 𝓣 on P, then for every open neighbourhood Up of p, there exists M0 βŠ‘ M such that β‹‚{↑m : m ∈ M0} βŠ† Up.

Definition 3.3

([8]). Let (P, π“œ) be a PM-space.

  1. For x, y ∈ P, define yβ‰ͺΞ±(π“œ)x if for every net (xi)i∈I that lim-infπ“œ-converges to x, xi β©Ύ y holds eventually.

  2. The poset P is said to be Ξ±(π“œ)-continuous if {x ∈ P : xβ‰ͺΞ±(π“œ)a} ∈ π“œ and a = sup{x ∈ P : xβ‰ͺΞ±(π“œ)a} holds for every a ∈ P.

Given a PM-space (P, π“œ), the approximate relation β‰ͺΞ±(π“œ) on the poset P can be equivalently characterized in the following proposition.

Proposition 3.4

Let (P, π“œ) be a PM-space and x, y ∈ P. Then yβ‰ͺΞ±(π“œ)x if and only if for every M ∈ π“œ with sup M β©Ύ x, there exists M0 βŠ‘ M such that

β‹‚{↑m:m∈M0}βŠ†β†‘y.

Proof

Suppose yβ‰ͺΞ±(π“œ)x. Let (x(d,D))(d,D)∈DMx be the net defined in Remark 3.2 (3) for every M ∈ π“œ with sup M = p β©Ύ x. Then the net (x(d,D))(d,D)∈DMxβ†’Mx. By Definition 3.3 (1), there exists (d0,D0) ∈ DMx such that x(d,D) = d β©Ύ y for all (d, D) ≦ (d0,D0). Since (d, D0) ≦ (d0,D0) for every d ∈ D0, x(d,D0) = d β©Ύ y for every d ∈ D0. So D0 βŠ† ↑y. This shows that there exists M0 βŠ‘ M such that D0 = β‹‚{↑m : m ∈ M0} βŠ† ↑y.

Conversely, suppose that for every M ∈ π“œ with sup M β©Ύ x, there exists M0 βŠ‘ M such that β‹‚{↑m : m ∈ M0} βŠ† ↑y. Let (xi)i∈I be a net that lim-infπ“œ-converges to x. Then, by Definition 3.1, there exists M ∈ π“œ such that sup M = p β©Ύ x, and for every m ∈ M, there exists im ∈ I such that xi β©Ύ m for all i β‰₯ im. Take i0 ∈ I with that i0 β‰₯ im for every m ∈ M0 βŠ‘ M, we have that xi ∈ β‹‚{↑m : m ∈ M0} βŠ† ↑y for all i β‰₯ i0. This shows that xi β©Ύ y holds eventually. Thus, by Definition 3.3 (1), we have yβ‰ͺΞ±(π“œ)x. ░

Remark 3.5

Let (P, π“œ) be a PM-space and x, y ∈ P.

  1. If there is no M ∈ π“œ such that sup M β©Ύ x, then pβ‰ͺΞ±(π“œ)x for every p ∈ P. And, if the poset P has the least element βŠ₯, then βŠ₯β‰ͺΞ±(π“œ)p for every p ∈ P.

  2. The implication yβ‰ͺΞ±(π“œ)x ⟹ y β©½ x may not be true. For example, let P = {0,1, 2, …} be in the discrete order β©½ defined by

    (βˆ€i,j∈P)iβ©½j⟺i=j.

    And let π“œ = {{2}}. Then, it is easy to see from Remark 3.5 (1) that 0β‰ͺΞ±(π“œ)1 and 0β§Έ β©½ 1.

  3. Assume the PM-space (P, π“œ) has the property that for every p ∈ P, there exists Mp ∈ π“œ such that sup Mp = p. Then, by Proposition 3.4, we have

    (βˆ€q,r∈P)qβ‰ͺΞ±(M)r⟹qβ©½r.

For more interpretations of the approximate relation β‰ͺΞ±(π“œ) on posets, the readers can refer to Example 3.2 and Remark 3.3 in [8].

For simplicity, given a PM-space (P, π“œ) and x ∈ P, we will denote

  • –

    β–Ύπ“œx = {y ∈ P : yβ‰ͺΞ±(π“œ)x};

  • –

    β–΄π“œx = {z ∈ P : xβ‰ͺΞ±(π“œ)z}.

Based on the approximate relation β‰ͺΞ±(π“œ) on posets, the Ξ±*(π“œ)-continuity can be defined for posets in the following:

Definition 3.6

Let (P, π“œ) be a PM-space. The poset P is called an Ξ±*(π“œ)-continuous poset if for every x ∈ P, there exists Mx ∈ π“œ such that

  • (O1)

    sup Mx = x and Mx βŠ† β–Ύπ“œx. And,

  • (O1)

    for every y ∈ β–Ύπ“œx, there exists F βŠ‘ Mx such that β‹‚{↑f : f ∈ F} βŠ† β–΄π“œy.

Noticing Remark 3.5 (3), we have the following proposition about Ξ±*(π“œ)-continuous posets.

Proposition 3.7

Let (P, π“œ) be a PM-space in which the poset P is Ξ±*(π“œ)-continuous. Then

(βˆ€x,y∈P)yβ‰ͺΞ±(M)x⟹yβ©½x.

The following examples of Ξ±*(π“œ)-continuous posets can be formally checked by Definition 3.6.

Example 3.8

Let (P, π“œ) be a PM-space.

  1. If P is a finite poset, then P is an Ξ±*(π“œ)-continuous poset if and only if for every x ∈ P, there exists Mx ∈ π“œ such that sup Mx = x.

  2. Let π“œ = 𝓛(P). Then P is an Ξ±*(𝓛)-continuous poset. This means that every poset is Ξ±*(𝓛)-continuous.

  3. Let π“œ = 𝓓(P). Then we have β‰ͺ = β‰ͺΞ±(𝓓) (see Example 3.2 (1) in [8]). The poset P is a continuous poset if and only if it is an Ξ±*(𝓓)-continuous poset. In particular, finite posets, chains, anti-chains and completely distributive lattices are all Ξ±*(𝓓)-continuous.

  4. Let π“œ = π“Ÿ(P). If P is a finite poset (resp. chain, anti-chain), then P is an Ξ±*(π“Ÿ)-continuous poset.

Proposition 3.9

Let (P, π“œ) be a PM-space. If P is an Ξ±(π“œ)-continuous poset, and {y ∈ P : (βˆƒ z ∈ P) yβ‰ͺΞ±(π“œ)zβ‰ͺΞ±(π“œ)a} ∈ π“œ for every a ∈ P, then P is an Ξ±*(π“œ)-continuous poset.

Proof

Suppose that P is an Ξ±(π“œ)-continuous poset, and {y ∈ P : (βˆƒ z ∈ P) yβ‰ͺΞ±(π“œ)zβ‰ͺΞ±(π“œ)a} ∈ π“œ for every a ∈ P. Take Ma = β–Ύπ“œa. Then it is easy to see that sup Ma = a and Ma βŠ† β–Ύπ“œa. By Remark 3.3 (4) in [8], we have sup{y ∈ P : (βˆƒ z ∈ P) yβ‰ͺΞ±(π“œ)zβ‰ͺΞ±(π“œ)a} = a. This implies, by Proposition 3.4 and Remark 3.5 (2), that for every y ∈ β–Ύπ“œa, there exist {y1,y2, …,yn}, {z1,z2, …,zn} βŠ‘ Ma = β–Ύπ“œa such that

β‹‚{↑zi:i∈{1,2,…,n}}βŠ†β‹‚{↑yi:i∈{1,2,…,n}}βŠ†β†‘y,

and yiβ‰ͺΞ±(π“œ)ziβ‰ͺΞ±(π“œ)a for every i ∈ {1, 2, …, n}. Next, we show β‹‚{↑zi : i ∈ {1, 2, …, n}} βŠ† β–΄π“œy. For every M ∈ π“œ with sup M β©Ύ b ∈ β‹‚{↑zi : i ∈ {1, 2, …, n}}, by Proposition 3.4, there exists Mi βŠ‘ M such that β‹‚{↑mβ€²:mβ€² ∈ Mi} βŠ† ↑yi for every i ∈ {1, 2, …, n}. Take M0 = ⋃{Mi : i ∈ {1, 2, …, n}}. Then M0 βŠ‘ M and

β‹‚{↑m:m∈M0}βŠ†β‹‚{↑yi:i∈{1,2,…,n}}βŠ†β†‘y.

This shows yβ‰ͺΞ±(π“œ)b for every b ∈ β‹‚{↑zi : i ∈ {1, 2, …, n}}. Hence, β‹‚{↑zi : i ∈ {1, 2, …, n}} βŠ† β–΄π“œy. Thus P is an Ξ±*(π“œ)-continuous poset. ░

The fact that an Ξ±*(π“œ)-continuous poset P in a PM-space (P, π“œ) may not be Ξ±(π“œ)-continuous can be demonstrated in the following example.

Example 3.10

Let (P, π“œ) be the PM-space in which the poset P = ℝ is the set of all real number with its usual order β©½ and π“œ = 𝓒0(ℝ). Then we have β‰ͺΞ±(𝓒0) = β©½ by Proposition 3.4. It is easy to check, by Definition 3.6, that ℝ is an Ξ±*(𝓒0)-continuous poset. But ℝ is not an Ξ±(𝓒0)-continuous poset because ▾𝓒0x = ↓xβ§Έ ∈ 𝓒0(P) for every x ∈ ℝ.

We turn to consider the topology induced by the lim-infπ“œ-convergence in posets.

Definition 3.11

Let (P, π“œ) be a PM-space. A subset V of P is said to be π“œ-open if for every net (xi)i∈Iβ†’Mx∈V, xi ∈ V holds eventually.

Given a PM-space (P, π“œ), one can formally verify that the set of all π“œ-open subsets of P forms a topology on P. This topology is called the π“œ-topology, and denoted by π“žπ“œ(P).

The following Theorem is an order-theoretical characterization of π“œ-open sets.

Theorem 3.12

Let (P, π“œ) be a PM-space. Then a subset V of P is π“œ-open if and only if it satisfies the following two conditions:

  • (V1)

    ↑V = V, i.e., V is an upper set.

  • (V2)

    For every M ∈ π“œ with sup M ∈ V, there exists M0 βŠ‘ M such that β‹‚{↑m : m ∈ M0} βŠ† V.

Proof

Suppose that V is an π“œ-open subset of P. By Remark 3.2 (1), it is easy to see that V is an upper set. Let (x(d,D))(d,D)∈DMx be the net defined in Remark 3.2 (3) for every M ∈ π“œ with sup M = x ∈ V. Then (x(d,D))(d,D)∈DMxβ†’Mx∈V. This implies, by Definition 3.11, that there exists (d0,D0) ∈ DMx such that x(d,D) = d ∈ V for all (d, D) β‰₯ q (d0,D0). Since (d, D0) ≦ (d0,D0) for all d ∈ D0, x(d,D0) = d ∈ V for all d ∈ D0. This shows D0 βŠ† V. Thus there exists M0 βŠ‘ M such that D0 = β‹‚{↑m : m ∈ M0} βŠ† V.

Conversely, suppose V is a subset of P which satisfies Condition (V1) and (V2). Let (xi)i∈I be a net that lim-infπ“œ-converges to x ∈ V. Then there exists M ∈ π“œ such that sup M = y β©Ύ x ∈ V = ↑V (hence, y ∈ V), and for every m ∈ M, there exists im ∈ I such that xi β©Ύ m for all i β‰₯ im. By Condition (V2), we have that β‹‚{↑m : m ∈ M0} βŠ† V for some M0 βŠ‘ M. Take i0 ∈ I with that i0 β‰₯ im for all m ∈ M0. Then xi ∈ β‹‚{↑m : m ∈ M0} βŠ† V for all i β‰₯ i0. This shows that V is an π“œ-open set. ░

Recall that given a topological space (X, 𝓣) and a point x ∈ P, a family 𝓑(x) of open neighbourhoods of x is called a base for the topological space (X, 𝓣) at the point x if for every neighbourhood V of x there exists an U ∈ 𝓑(x) such that x ∈ U βŠ† V.

If the poset P in a PM-space (P, π“œ) is an Ξ±*(π“œ)-continuous poset, we provide a base for the topological space (P, π“žπ“œ(P)) at a point x ∈ P.

Proposition 3.13

Let (P, π“œ) be a PM-space in which the poset P is Ξ±*(π“œ)-continuous. Then β–΄π“œx ∈ π“žπ“œ(P) for every x ∈ P.

Proof

One can readily see, by Proposition 3.4, that {β–΄π“œ}x is an upper subset of P for every x ∈ P. For every M ∈ π“œ with sup M = y ∈ {β–΄π“œ}x, by Definition 3.6 (O1) there exists My ∈ π“œ such that My βŠ† {β–Ύπ“œ}y and sup My = y. Since xβ‰ͺΞ±(π“œ)y, by Definition 3.6 (O2), we have β‹‚{↑mi : i ∈ {1, 2, …, n}} βŠ† {β–΄π“œ}x for some {m1,m2, …,mn} βŠ‘ My. Observing {m1,m2, …,mn} βŠ‘ My βŠ† {β–Ύπ“œ}y, we can conclude that there exists Mi βŠ‘ M such that β‹‚{↑a : a ∈ Mi} βŠ† ↑mi for every i ∈ {1, 2, …, n}. Let M0 = ⋃{Mi : i ∈ {1, 2, …, n}}. Then M0 βŠ‘ M and

β‹‚{↑m:m∈M0}βŠ†β‹‚{↑mi:i∈{1,2,…,n}}βŠ†β–΄Mx.

This shows, by Theorem 3.12, that β–΄π“œx ∈ π“žπ“œ(P) for every x ∈ P. ░

Proposition 3.14

Let (P, π“œ) be a PM-space in which the poset P is Ξ±*(π“œ)-continuous and x ∈ P. Then {β‹‚{β–΄π“œa : a ∈ A} : A βŠ‘ β–Ύπ“œx} is a base for the topological space (P, π“žπ“œ(P)) at the point x.

Proof

Clearly, by Proposition 3.13, we have β‹‚{β–΄π“œa : a ∈ A} ∈ π“žπ“œ(P) for every A βŠ‘ β–Ύπ“œx. Let U ∈ π“žπ“œ(P) and x ∈ U. Since P is an Ξ±*(π“œ)-continuous poset, there exists Mx ∈ π“œ such that Mx βŠ† β–Ύπ“œx and sup Mx = x ∈ U. By Theorem 3.12, it follows that β‹‚{↑m : m ∈ M0} βŠ† U for some M0 βŠ‘ Mx βŠ† β–Ύπ“œx. So, from Proposition 3.7, we have

xβˆˆβ‹‚{β–΄Mm:m∈M0}βŠ†β‹‚{↑m:m∈M0}βŠ†U.

Thus, {β‹‚{β–΄π“œa : a ∈ A} : A βŠ‘ β–Ύπ“œx} is a base for the topological space (P, π“žπ“œ(P)) at the point x. ░

In the rest, we are going to establish a characterization theorem which demonstrates the equivalence between the lim-infπ“œ-convergence being topological and the Ξ±*(π“œ)-continuity of the poset in a given PM-space.

Lemma 3.15

Let (P, π“œ) be a PM-space. If P is an Ξ±*(π“œ)-continuous poset, then a net

(xi)i∈Iβ†’Mx∈P⟺(xi)i∈Iβ†’OM(P)x.

Proof

By the definition of π“žπ“œ(P), it is easy to see that a net

(xi)i∈Iβ†’Mx∈P⟹(xi)i∈Iβ†’OM(P)x.

To prove the Lemma, we only need to show that a net (xi)i∈Iβ†’OM(P)x∈P implies (xi)i∈Iβ†’Mx. Suppose (xi)i∈Iβ†’OM(P)x. As P is an Ξ±*(π“œ)-continuous poset, there exists Mx ∈ π“œ such that Mx βŠ† β–Ύπ“œx and sup Mx = x. By Proposition 3.13, we have x ∈ β–΄π“œy ∈ {π“žπ“œ(P)} for every y ∈ Mx βŠ† β–Ύπ“œx. Hence, xi ∈ β–΄π“œy holds eventually. This implies, by Proposition 3.7, that xi ∈ β–΄π“œy βŠ† ↑y holds eventually. By the definition of lim-infπ“œ-convergence, we have (xi)i∈Iβ†’Mx. ░

In the converse direction, we have the following Lemma.

Lemma 3.16

Let (P, π“œ) be a PM-space. If the lim-infπ“œ-convergence in P is topological, then P is an Ξ±*(π“œ)-continuous poset.

Proof

Suppose that the lim-infπ“œ-convergence in P is topological. Then there exists a topology 𝓣 such that for every x ∈ P, a net

(xi)i∈Iβ†’Mx⟺(xi)i∈Iβ†’Tx.

Define Ix = {(p, V) ∈ P Γ— 𝓝(x) : p ∈ V}, where 𝓝(x) is the set of all open neighbourhoods of x, namely, 𝓝(x) = {V ∈ 𝓣 : x ∈ V}. Define also the preorder βͺ― on Ix as follows:

(βˆ€(p1,V1),(p2,V2)∈Ix)(p1,V1)βͺ―(p2,V2)⟺V2βŠ†V1.

It is easy to see that Ix is directed. Now, let x(p,V) = p for every (p, V) ∈ Ix. Then one can readily check that the net (x(p,V))(p,V)∈Ixβ†’Tx, and hence (x(p,V))(p,V)∈Ixβ†’Mx. This means that there exists Mx ∈ π“œ such that sup Mx β©Ύ x, and for every m ∈ Mx, there exists (pm,Vm) ∈ Ix with that x(p,V) = p β©Ύ m for all (p, V) βͺ° (pm,Vm). Since (p, Vm) βͺ° (pm,Vm) for all p ∈ Vm, we have x(p,Vm) = p β©Ύ m for all p ∈ Vm. This shows

(βˆ€m∈Mx)(βˆƒVm∈N(x))x∈VmβŠ†β†‘m.(⋆⋆)

Next we prove Mx βŠ† β–Ύπ“œx. For every m ∈ Mx and every M ∈ M with sup M β©Ύ x, let (x(d,D))(d,D)∈DMx be the net defined in Remark 3.2 (3). Then the net (x(d,D))(d,D)∈DMxβ†’Mx, and thus (x(d,D))(d,D)∈DMxβ†’Tx. It follows from Remark 3.2 (4) that there exists M0 βŠ‘ M such that x ∈ β‹‚{↑a : a ∈ M0} βŠ† Vm. By Condition (⋆⋆), we have x ∈ β‹‚{↑a : a ∈ M0} βŠ† Vm βŠ† ↑m. So, mβ‰ͺΞ±(π“œ)x. This shows Mx βŠ† β–Ύπ“œx.

Let y ∈ β–Ύπ“œx. Then there exists {m1,m2, …,mn} βŠ‘ Mx such that β‹‚{↑mi : i ∈ {1, 2, …, n}} βŠ† ↑y as Mx ∈ π“œ and sup Mx β©Ύ x. By Condition (⋆⋆), it follows that β‹‚Vmi : i ∈ {1, 2, …, n}} βŠ† β‹‚{↑mi : i ∈ {1, 2, …, n}} βŠ† ↑y. Considering the net (x(d,D))(d,D)∈DMxx defined in Remark 3.2 (3), we have (x(d,D))(d,D)∈DMxxβ†’M x, and hence (x(d,D))(d,D)∈DMxxβ†’Tx. This implies, by Remark 3.2 (4), that

β‹‚{↑b:b∈M00}βŠ†β‹‚{Vmi:i∈{1,2,…,n}}βŠ†β‹‚{↑mi:i∈{1,2,…,n}}βŠ†β†‘y(⋆⋆⋆)

for some M00 βŠ‘ Mx. Finally, we show β‹‚{↑b : b ∈ M00} βŠ† β–΄π“œy. For every xβ€² ∈ β‹‚{↑b : b ∈ M00} and every Mβ€² ∈ π“œ with sup Mβ€² β©Ύ xβ€², let (x(d,D))(d,D)∈DMβ€²xβ€² be the net defined in Remark 3.2 (3). Then (x(d,D))(d,D)∈DMβ€²xβ€²β†’Mxβ€², and thus (x(d,D))(d,D)∈DMβ€²xβ€²β†’Txβ€². It follows from Condition (⋆⋆⋆) and Remark 3.2 (4) that there exists M0β€²βŠ‘Mβ€² such that

β‹‚{↑aβ€²:aβ€²βˆˆM0β€²}βŠ†β‹‚{Vmi:i∈{1,2,…,n}}βŠ†β‹‚{↑mi:i∈{1,2,…,n}}βŠ†β†‘y.

This shows xβ€² ∈ β–΄π“œy, and thus β‹‚{↑b : b ∈ M00} βŠ† β–΄π“œy. Therefore, P is an Ξ±*(π“œ)-continuous poset. ░

Combining Lemma 3.15 and Lemma 3.16, we deduce the following result.

Theorem 3.17

Let (P, π“œ) be a PM-space. The following statements are equivalent:

  1. P is an Ξ±*(π“œ)-continuous poset.

  2. For any net (xi)i∈I in P,(xi)i∈Iβ†’Mx∈P⟺(xi)i∈Iβ†’OM(P)x.

  3. The lim-infπ“œ-convergence in P is topological.

Proof

(1) β‡’ (2): By Lemma 3.15.

(2) β‡’ (3): Clear.

(3) β‡’ (1): By Lemma 3.16.β–‘

Corollary 3.18

([8]). Let (P, π“œ) be a PM-space with 𝓒0(P) βŠ† π“œ βŠ† π“Ÿ(P). Suppose β–Ύπ“œa ∈ π“œ and {y ∈ P : (βˆƒ z ∈ P) yβ‰ͺΞ±(π“œ)zβ‰ͺΞ±(π“œ)a} ∈ π“œ holds for every a ∈ P. Then the lim-infπ“œ-convergence in P is topological if and only if P is Ξ±(π“œ)-continuous.

Proof

(⟹): To show the Ξ±(π“œ)-continuity of P, it suffices to prove supβ–Ύπ“œa = a for every a ∈ P. Since the lim-infπ“œ-convergence in P is topological, by Theorem 3.17, P is an Ξ±*(π“œ)-continuous poset. This implies that there exists Ma ∈ π“œ such that sup Ma βŠ† β–Ύπ“œa and sup Ma = a for every a ∈ P. By Proposition 3.7, we have β–Ύπ“œa βŠ† ↓a. So sup β–Ύπ“œa = a.

(⟸): By Proposition 3.9 and Theorem 3.17.β–‘

Acknowledgement

This work is supported by the Doctoral Scientific Research Foundation of Hunan University of Arts and Science (Grant No.: E07017024), the Significant Research and Development Project of Hunan province (Grant No.: 2016JC2014) and the Natural Science Foundation of China (Grant No.: 11371130).

References

  • [1]

    Birkhoff G., Lattice Theory,1940, American Mathematical Society Colloquium Publications.Β GoogleΒ Scholar

  • [2]

    Frink O., Topology in lattice, Trans. Am. Math. Soc., 1942, 51, 569-582.Β CrossrefGoogleΒ Scholar

  • [3]

    Mcshane E.J., Order-Preserving Maps and Integration Process, 1953, Princeton, Princeton University Press.Β GoogleΒ Scholar

  • [4]

    Zhao B., Zhao D.S., Lim-inf-convergence in partially ordered sets, J. Math. Anal. Appl., 2005, 309, 701-708.Β CrossrefGoogleΒ Scholar

  • [5]

    Wang K.Y., Zhao B., Some further results on order-convergence in posets, Topol. Appl., 2013, 160, 82-86.Β WebΒ ofΒ ScienceCrossrefGoogleΒ Scholar

  • [6]

    Zhao B., Li J., O2-convergence in posets, Topol. Appl., 2006, 153, 2971-2975.Β CrossrefGoogleΒ Scholar

  • [7]

    Li Q.G., Zou Z.Z., A result for O2-convergence to be topological in posets, Open Math., 2016, 14, 205-211.Β WebΒ ofΒ ScienceGoogleΒ Scholar

  • [8]

    Zhou Y.H., Zhao B., Order-convergence and lim-infπ“œ-convergence in posets, J. Math. Anal. Appl., 2007, 325, 655-664.Β WebΒ ofΒ ScienceCrossrefGoogleΒ Scholar

  • [9]

    Venugopalan P., Z-continuous posets, Houston J. Math., 1996, 12, 275-294.Β GoogleΒ Scholar

  • [10]

    Kelly J.L., General Topology, 1955, New York, Van Nostrand Group.GoogleΒ Scholar

  • [11]

    Mathews J.C., Anderson R.F., A comparison of two modes of order convergence, Proc. Am. Math. Soc., 1967, 18(1), 100-104.Β CrossrefGoogleΒ Scholar

  • [12]

    Wolk E.S., On order-convergence, Proc. Am. Math. Soc., 1961, 12(3), 379-384.CrossrefGoogleΒ Scholar

  • [13]

    Zhao D.S., The double Scott topology on a lattice, Chin. Ann. Math., Ser. A., 1989, 10(2), 187-193.GoogleΒ Scholar

  • [14]

    Zhao B., Wang K.Y., Order topology and bi-Scott topology on a poset, Acta Math. Sin. Engl. Ser., 2011, 27, 2101-2106.CrossrefGoogleΒ Scholar

  • [15]

    Engelking R., General Topology, 1977, Warszawa, Polish Scientific Publishers.GoogleΒ Scholar

  • [16]

    Zhang H., A note on continuous partially ordered sets, Semigroup Forum, 1993, 47, 101-104.CrossrefGoogleΒ Scholar

  • [17]

    Davey B.A., Priestley H.A., Introduction to Lattices and Order, 2002, Cambridge, Cambridge University Press.GoogleΒ Scholar

  • [18]

    Grierz G., Hofman K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S., Continuous Lattices and Domain, 2003, Cambridge, Camberidge University Press.GoogleΒ Scholar

  • [19]

    Grierz G., Hofman K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S., A Compendium of Continuous Lattices, 1980, Berlin: Springer-Verlag Press.GoogleΒ Scholar

  • [20]

    Olejček V., Order convergence and order topology on a poset, Int. J. Theor. Phys., 1999, 38, 557-561.CrossrefGoogle Scholar

Footnotes

  • 1

    From the logical point of view, we stipulate β‹‚{↑a ∩ ↓b : a ∈ A0 & b ∈ B0} = P if A0 = βˆ… or B0 = βˆ….Β 

  • 2

    From the logical point of view, we stipulate β‹‚{↑m : m ∈ M0} = P if M0 = βˆ….Β 

About the article

Received: 2018-04-04

Accepted: 2018-07-17

Published Online: 2018-09-18


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1077–1090, ISSN (Online) 2391-5455, DOI:Β https://doi.org/10.1515/math-2018-0090.

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Β© 2018 Sun et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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